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A185342
Triangle of successive recurrences in columns of A117317(n).
1
2, 4, -4, 6, -12, 8, 8, -24, 32, -16, 10, -40, 80, -80, 32, 12, -60, 160, -240, 192, -64, 14, -84, 280, -560, 672, -448, 128, 16, -112, 448, -1120, 1792, -1792, 1024, -256, 18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512, 20, -180, 960, -3360, 8064
OFFSET
0,1
COMMENTS
A117317 (A):
1
2 1
4 5 1
8 16 9 1
16 44 41 14 1
32 112 146 85 20 1
64 272 456 377 155 27 1
have for their columns successive signatures
(2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).
Take based on abs(A133156) (B):
1
2 0
4 1 0
8 4 0 0
16 12 1 0 0
32 32 6 0 0 0
64 80 24 1 0 0 0.
The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.
FORMULA
Take A133156(n) without 1's or -1's double triangle (C)=
2
4
8 -4
16 -12
32 -32 6
64 -80 24
128 -192 80 -8
256 -448 240 -40
512 -1024 672 -160 10;
a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).
Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.
T(n,k) = T(n-1,k) - 2*T(n-1,k-1) for k>1, T(n,1) = 2*n = 2*T(n-1,1) - T(n-2,1). - Philippe Deléham, Feb 11 2012
T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - Philippe Deléham, Feb 11 2012
EXAMPLE
Triangle T(n,k),for 1<=k<=n, begins :
2 (1)
4 -4 (2)
6 -12 8 (3)
8 -24 32 -16 (4)
10 -40 80 -80 32 (5)
12 -60 160 -240 192 -64 (6)
14 -84 280 -560 672 -448 128 (7)
16 -112 448 -1120 1792 -1792 1024 -256 (8)
Successive rows can be divided by A171977.
MATHEMATICA
Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 27 2017 *)
PROG
(PARI) for(n=1, 20, for(k=1, n, print1((-2)^(k+1)*binomial(n, k)/2, ", "))) \\ G. C. Greubel, Jun 27 2017
CROSSREFS
Cf. For (A): A053220, A056243. For (B): A000079, A001787, A001788, A001789. For A193862: A115068 (a Coxeter group). For (2): A014480 (also (3),(4),(5),..); also A053220 and A001787.
Cf. A007318.
Sequence in context: A022471 A359294 A224487 * A276985 A081238 A333823
KEYWORD
sign,tabl
AUTHOR
Paul Curtz, Jan 26 2012
STATUS
approved